There are several conflicts in Edgar Allan Poe's short story "The Tell-Tale Heart." Let's examine them:
The first conflict is both external and internal. The narrator begins the story by asking the reader, "But why will you say that I am mad?" This conflict is external in that the narrator is attempting to convince his audience that he is only ill and not crazy; it is internal in that the narrator is also trying to convince himself of this same thing! This conflict is resolved through the act of telling the story of his madness.
The next conflict is a bit more obvious: the external conflict between the old man, who has the eye of a "vulture," and the narrator, who wishes to kill him. The narrator sneaks into the old man's room every night in order to "plan" (which arguably could be viewed as internal conflict, as well; i.e. "How do I kill him? When?"). This conflict is ultimately resolved when the narrator murders the old man.
The final conflict is, once again, both external and internal. The narrator has the external threat of police officers coming to check up on the household after hearing the old man's cry, which poses a certain risk that they might discover the dead body. The internal aspect of this conflict is the persistent beating of the dead heart that the narrator imagines he is hearing from beneath the floorboards. This conflict is resolved by the narrator confessing to the murder since he can no longer stand to listen to the heartbeat.